Asymptiotics of the systematic error in stochastic homogenization

We study a uniformly elliptic equation with high oscillatory (at scale S<1) and random coefficient. It is well known since the 70’ with the work of Kozlov, Papanicolaou and Varadhan that the solution, provided the law of the coefficient is stationary and ergodic, can be approximated by its two-scale expansion: that is a first-order approximation in S taking account of the oscillation at scale S. The zero-order term corresponds to the limit as S goes to zero and solve a nice elliptic PDE with deterministic constant coefficient (called homogenized coefficient) for which we have an explicit formula. However, the homogenized coefficient depends on the so-called first-order corrector which solves an elliptic PDE posed in the whole space and for which numerical computations are out of reach. The goal of this work is to analyse the approximation of the homogenized coefficient by the representative volume element method, that is when we replace the whole space by a torus of size L>>1. This operation makes the corrector equation easy to solve numerically and provide a natural approximation of the homogenized coefficient by the one coming from periodic homogenization. The variance of the difference suffers from two types of error: a random (that is fluctuation around its expectation) and a systematic one. We focus in this work on the systematic error and we characterize its asymptotic behaviour as L goes to infinity. We show, in the particular case where the law is generated by a stationary Gaussian field, that the asymptote is characterized by a deterministic matrix depending on the first and second-order correctors, the gradient of the covariance function and a fourth-order tensor involving the whole space Green function of the homogenized elliptic operator.